Find the maximum number of regions that $n$ circles divide a plane into.
Let $C_{n}$ be the maximum number of regions in which $n$ circles divide a plane. Clearly, $ C_{1} = 2, C_{2} =4, C_{3}=8 $ but $C_{4}=14,$ whoops!
The number of regions will be maximum if every two circles intersect in two distinct points and no three circles intersect in one point. Suppose that there are $(r-1)$ circles in a plane and they have divided the plane into $C_{r-1}$ regions. Every time the $r$th circle intersects another circle, it splits the original region into two, thereby adding one more region. Since the $r$th circle will intersect each of the $(r-1)$ circles in $2$ distinct points, therefore, it will add $2(r-1)$ regions. Therefore, we have the following recurrence
Comment: Quite often, sets are represented by circles in Venn diagrams. This result clearly shows that $n$ circles cannot form an $n$-set Venn diagram for $n\ge4$. Indeed, $n$ sets have $2^n$ possible membership patterns, whereas $n$ circles can divide the plane into at most $n^2-n+2$ regions. Since
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